I will be honest - I do have a bias against gymnastics environment that drives girls towards competitive gymnastics after seeing the regrets of my co-worker with his daughter where he regretted not having her quit sooner (the carousel of injuries and stress got too much) so I can not speak to specifics of gymnastics.
Some sports are such that individual coaching works a lot better than others (as some people mentioned) but even in sports like soccer and volleyball, there is still some benefit to periodic individual coaching especially for position specific skills like goalkeeping in soccer or setter in volleyball.
There is a difference between raw physical skill and coach-ability of an individual - some people are able to watch other people and with basic coaching, seem to leap in bounds with what they can do while others take longer to get to that same level but may have more potential ultimately to go further. Some sports will be easier than others for those who require more time to develop... I personally have been a good athlete but it takes me longer to make the adjustments I need to technique. But once I get that down, I catch up or surpass my class or teammates... It was like that for me in soccer, horseback riding, volleyball, rowing... I never started out in the top group but usually would in a year or season, surprise my coach with what my capability ended up being relative to my teammates.
I am very visual person, and found benefit in being in group situations by watching what others did right and wrong (but this is a maturity issue) but improved best when a coach worked with me personally. That is because although visually, I could see and understand the changes I needed to make, I could not feel the differences without feedback.
Music... Individual lessons are ideal, from my experience and both of my sisters (one who did both singing and violin at very high levels through high school).
